What do we see characterizing the Muslim prophet that is an improvement on the teachings, leadership and example of Moses?
Your question is in error. Read the 2nd to last aya in sura 2 of Quran (Al Baqara)(2:285). Where it says "We (Muslims) believe in one God, and his angels, his scriptures, and his prophets. We make no distinction between his prophets "
So Muslims cannot say that any prophet is 'better' than another. These comparisons are based on egoism, which is a human weakness. Prophets are way above the rest of humanity in this regard.
Firstly, this question appears to b a "my Prophet is better than yours" kind of issue, which in Islam is inherently rejected on the basis that they were all together as one brotherhood, on the same mission, sent by the same One True God, so just to be very clear, in that respect, they are brothers to one another and there is no animosity or competition within them in that sense. In this context, and perhaps related to the famous incident of Yunus (Jonah):
عَنْ أَبِي هُرَيْرَةَ عَنْ النَّبِيِّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ قَالَ لَا يَنْبَغِي لِعَبْدٍ أَنْ يَقُولَ أَنَا خَيْرٌ مِنْ يُونُسَ بْنِ مَتَّى - وفي رواية قال مَنْ قَالَ أَنَا خَيْرٌ مِنْ يُونُسَ بْنِ مَتَّى فَقَدْ كَذَبَ Abu Huraira reported: The Prophet, peace and blessings be upon him, said, “No servant should say that I am better than Jonah son of Mata.” In another narration, the Prophet said, “Whoever says that I am better than Jonah son of Mata has told a lie.” (Bukhari and Muslim)
The Answer > After this, the answer is that the superiority of the Prophet has been established by God, in many respects, over other Prophets. Of them are:
❶ - Muhammad was sent for the whole world, not just to a particular nation. This is clear in the respective scriptures, which limit Moses and Jesus to the children of Israel, and explicitly mention the Prophet Muhammad as being sent for the whole world.
❷ - The scriptures before Muhammad prophesied him and acknowledged him as being the one to follow when he arrives. In other words, the previous people and the previous Prophets were commanded to follow Muhammad and that is clear superiority.
❸ - The Qur'an is the most powerful & timeless miracle of God (far eclipsing other miracles which were limited in time and scope). The Qur'an is a miracle which is constant until the end of time, which can be witnessed by whoever wishes at any time. (This may be difficult to understand without explaining or learning what true miracles are, but this is neither the time nor the place for that discussion).
These 3 are sufficient to prove the point here. However, I am sure you will find extensive scholarly volumes written on this topic.
Muslims don't regard as prophet Muhammad to be better. This is mentioned in Quran, Al-Baqarah: 285
The Messenger has believed in the Guidance which has been sent down to him from his Lord, and those who believe in the Messenger have also sincerely accepted the same. They all believe in Allah, His Angels, His Books and Messengers. And they say, "We do not discriminate against any of His Messengers. We have heard the Message and submitted to it. Our Lord, we look up to You for forgiveness, for to You we shall all return
Hence we also admit Moses as one of the prophet and messenger hence there are virtues that we could follow from him. An example would be Quran, Al-Kahf: 60-82, where the quran tells the story of prophet Moses and prophet Khidr.
Muslims does follow prophet Muhammad though in teaching for the sole reason that prophet Muhammad is the final prophet that delivers the complete teaching of Islam. Hence, why it could be misunderstood that Muhammad is regarded higher as though it was not the intention.